Consider an unknown number of adult deer, $N$, which can be split into two groups - males, $M$, and females, $F$, where $N = F + M$. These adults were hunted during the year (a lethal form of sampling without replacement), with each group having its own individual-level probability of being killed by a hunter, $p_f$ and $p_m$, respectively. For example, male deer in Michigan have a 40% of being killed by a hunter during the hunting season while females only have a 16% chance of being killed. The number of adults that were killed can be denoted as $m$ and $f$, respectively.
I want to build a likelihood for the total number of adults, $N$, given that I know $p_m$, $p_f$, $m$, and $f$, to be used in a larger joint likelihood. If I were interested in building a likelihood for the two different groups of adults, I could write: $$L(M \mid p_m,m) = \binom{M}{m} \cdot (p_m)^{m} \cdot (1-p_m)^{M-m}$$ $$L(F \mid p_f,f) = \binom{F}{f} \cdot (p_f)^{f} \cdot (1-p_f)^{F-f}$$ What I want, however, is $$L(N \mid p_m,p_f,m,f) = \text{?}$$ The full likelihood already has a large number of parameters to estimate, so I'd rather just have a single parameter for the total number of adults, as opposed to having a separate parameters to estimate for males and females. Basically, the likelihood equation cannot contain $M$ or $F$.
Any help would be greatly appreciated!